The number of points, where the curve y=x^5-20x^3+50x+2 crosses the x-axis, is ___________…

Question

The number of points, where the curve $y={x}^{5}-20{x}^{3}+50x+2$ crosses the $x$-axis, is \_\_\_\_\_\_\_\_\_\_\_\_.

Answer: 5

Accepted Answer

Given equation of curve

$\begin{gathered} y={x}^{5}-20{x}^{3}+50x+2 \ \Rightarrow \frac{dy}{dx}=5{x}^{4}-60{x}^{2}+50 \end{gathered}$

On putting $\frac{dy}{dx}=0$

$\begin{gathered} \Rightarrow 5({x}^{4}-12{x}^{2}+10)=0 \ \ \Rightarrow {x}^{2}=\frac{12\pm \sqrt{144-40}}{2}=6\pm \sqrt{26} \ \ \Rightarrow {x}^{2}=6-\sqrt{26},6+\sqrt{26} \ \ \Rightarrow {x}^{2}=6-5.10,6+5.10 \ \ \Rightarrow {x}^{2}=09,11.1 \ \ \Rightarrow x=\pm \sqrt{0.9},\pm \sqrt{11.1} \ \ \Rightarrow x=-0.95,0.95,-3.33,3.33 $

Now,

$ y(0) =2(+ve)\Rightarrow y(1)=+ve \ \ y(2) =-ve\Rightarrow y(3.3)=-ve \ \ y(-1) =-ve\Rightarrow y(-2)=+ve \ \ y(-3.3) =-ve \end{gathered}$

$\because =5$

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